The definition of Berry curvature here is the same as that of Ref. [1]}, that is,
\(B _{k_x k_y }^\sigma =\frac{\partial }{{\partial k_x }}{\bf A}_{k_y }^\sigma - \frac{\partial }{{\partial k_y }}{\bf A}_{k_x }^\sigma \) .
From the above derivation, we find that a Berry curvature can be defined
uniquely for each band by looking at the phases of the polarized vectors
of both the displacements and conjugate momenta as functions of the wave vector.
If we only look at the polarized vector \(\epsilon \)  of the displacement, a
Berry curvature cannot properly be defined. We need both \(\epsilon \)  and \(\mu \) .
The nontrivial Berry vector potential takes the responsibility of the PHE.
The associated topological Chern number is obtained through integrating
the Berry curvature over the first Brillouin zone as
\(C^\sigma = \frac{1}{{2\pi }}\int _{{\rm BZ}} {dk_x dk_y B_{k_x k_y }^\sigma }= \frac{{2\pi }}{{L^2}}\sum \limits _{\bf k}{B _{k_x k_y }^\sigma },\) 
